If f(x)= array ll a x+3, & x 2 a² x-1 & x>2 array . , then the values of a for which f is continuous for all x are

Step-by-step solution

For the function f(x) to be continuous for all x , it must be continuous at the point x = 2 . The condition for continuity at x = 2 is _ x 2⁻ f(x) = _ x 2⁺ f(x) = f(2) . Calculating the left-hand limit at x = 2 : _ x 2⁻ f(x) = _ x 2⁻ (ax + 3) = 2a + 3 . Calculating the right-hand limit at x = 2 : _ x 2⁺ f(x) = _ x 2⁺ (a²x - 1) = 2a² - 1 . Equating the two limits: 2a + 3 = 2a² - 1 . Rearranging the equation: 2a² - 2a - 4 = 0 . Dividing by 2: a² - a - 2 = 0 . Factoring the quadratic equation: (a - 2)(a + 1) = 0 . The